Bring a large pot of well-salted water to a boil. Cook penne according to package directions until al dente. During the last 3 minutes of cooking time, drop broccoli florets directly into the pasta water.

Before draining, scoop out 1 cup of starchy pasta water and set aside in a measuring cup. Drain pasta and broccoli together and do not rinse.

Heat olive oil and 2 tbsp butter together in your largest skillet over medium-low heat. Add diced onion and cook for 4 to 5 minutes until soft and translucent. Add minced garlic and stir constantly for 30 seconds until fragrant.

Pour in heavy cream and chicken broth. Bring to a gentle simmer for 2 to 3 minutes, looking for small bubbles at the edges rather than a rolling boil.

Remove the skillet completely from heat before adding any cheese. Whisk in Parmesan and mozzarella in small amounts, stirring until each addition melts fully. Season with Italian seasoning, garlic powder, salt, pepper, and red pepper flakes if using.

Add drained pasta and broccoli to the sauce and toss gently with tongs until everything is evenly coated. Fold in shredded rotisserie chicken last.

Add reserved pasta water 2 tablespoons at a time, stirring between each addition, until the sauce is creamy and glossy. Stir in a cold knob of butter just before serving.